The most exotic among these integrable hierarchies
will be the equivariant Toda hierarchy.
This integrable hierarchy, like the 1D Toda hierarchy,
is a reduction of the 2D Toda hierarchy.
Okounkov and Pandharipande [1]}
encode the Gromov-Witten invariants
into the vacuum expectation value of an operator product
on the fermionic Fock space of 2D charged free fermion fields.
This fermionic expression can be converted
into a tau function of the 2D Toda hierarchy,
and eventually turns out to be a tau function
of the equivariant Toda hierarchy.
Milanov [2]} starts from a bosonic expression
of the Gromov-Witten invariants
in the Givental theory [3]},
and derives the Hirota bilinear equations
of the equivariant Toda hierarchy.
These results are further generalized to \(\mathbb {C}\mathbb {P}^1\) 
with two orbifold points [4]}, [5]}.
